Let $x, m, a, s$ be positive real numbers such that $x, m, a, s < 1$. Prove that $$\frac{x^4}{m^3 + a^2 + s} +\frac{m^4}{a^3 + s^2 + x} + \frac{a^4}{s^3 + x^2 + m} + \frac{s^4}{x^3 + m^2 + a} > \frac{x^3 + m^3 + a^3 + s^3}{3}$$.

Santa decorates his Christmas tree with a triangular decoration. Suppose the triangular decoration can be represented by an acute-angled scalene triangle $\vartriangle ABC$ with circumcircle $\omega$. $D$ is the foot of $B$-altitude of $\vartriangle ABC$. $E$ and $F$ are the points of projection of $D$ on $AB$ and $BC$ respectively. $A'$ is the point of reflection of $A$ over $BC$. $AA'$ meets $\omega$ again at $T$. $TD$ meets $\omega$ again at $X$. $Y$ is a point on $\omega$ such that $\angle TY A' = 90^o$. Prove that$ AY$ , $BX$ and $EF$ concur.

After delivering all the Christmas presents, Santa finally have some leisure time to study Mathematics, which is his favourite hobby. Santa defines a positive integer n as a “Christmas number” if $n$ is a divisor of $\phi (n) \sigma (n) + 1$. Santa challenges Rudolph the red nosed reindeer and asks him to find two prime numbers such that their product is a “Christmas number”. Is it possible for Rudolph to do so? Prove your claim. Note: $\sigma (n)$ denotes the sum of positive divisors of $n$. $\phi (n)$ denotes the Euler’s totient function of $n$, which counts the positive integers up to $n$ that are relatively prime to $n$.

We say that a non-empty set $A \subset Z$ is weird if and only if $$\left( \sum_{a \in A} a - |A| \right) \in A.$$Santa has $n$ elves, where $n$ is a fixed positive integer greater than or equal to $2$. Santa assigns each one of his $n$ elves a positive integer from $1$ up to $n$. Santa wants to ask some of the elves to create toys but he notices that if the set of elves that works on creating toys has a weird subset, they become lazy and inefficient. What is the maximal set (in terms of cardinality) of elves that can work simultaneously on crafting toys and stay efficient at the same time?

A positive integer is a Zumkeller number if its positive divisors can be partitioned into two disjoint sets with the same sum. For example, $6$ is a Zumkeller number because we can partition the positive divisors of $6$ into two disjoint sets $\{1, 2, 3\}$ and $\{6\}$, and the sum of elements of the two sets are equal. Santa defines a ridiculous number $R_n$ to be a number of the form $$R_n = m^2 - 69m - (n^2 - 69n)$$such that $R_n$ is also a Zumkeller number ($m$ and $n$ are positive integers). Santa is lazy this year, so he decides to fix $n \in N$ and give as many gifts as there are ridiculous numbers $R_n$. Prove that Santa’s plan is flawed as for any $n \in N$, there are infinitely many ridiculous numbers $R_n$.

Santa gives his elves a task. He gives them a square paper, denoted as $ABCD$, with sidelength $\ell$. Santa has marked a point $E$ on segment $AB$ such that $AE = x$, where $x < \frac{\ell}{6}$ . Santa defines a “Christmas pentagon” as a pentagon where $4$ of the sides are tangent to a single circle and Santa calls the radius of this circle the “Christmas radius” of the pentagon. Santa asks his elves to construct the following figures by folding the paper, without other construction instruments: (1) a “Christmas pentagon” AND (2) a triangle with inradius $x$ and circumradius which is equal to the “Christmas radius” of the “Christmas pentagon” in (1). If the elves can do so, they can get an extra Chrismas gift from santa, which is a cute christmas frog. Help the elves to complete their task and prove that your method works.